Supersingular locus of Hilbert modular varieties, arithmetic level raising and Selmer groups

TL;DR

This paper extends classical results on supersingular loci and level raising from modular curves to higher-dimensional Shimura varieties, and applies these to study Selmer groups related to the Bloch--Kato conjecture.

Contribution

It generalizes the characterization of supersingular loci and level raising to all Shimura varieties from quaternion algebras over totally real fields, and applies these to Selmer group analysis.

Findings

Generalized supersingular locus characterization to all Shimura varieties.

Extended arithmetic level raising results to inert cases.

Applied results to study Selmer groups in the context of the Bloch--Kato conjecture.

Abstract

This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.